#346030

order	512
nilpotency class	3
isoclinism	isoclinic class of #345968
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3205	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

509 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,8)(4,12)(5,7)(6,9)(10,16)(11,14)(13,15)(17,18,21,27)(19,24,33,42)(20,34,35,44)(22,29,23,28)(25,38,26,36)(30,40,47,43)(31,48,46,39)(32,45,37,41);(2,7)(3,10)(4,6)(8,15)(11,13)(14,16)(17,19,23,32,21,33,22,37)(18,24,29,41,27,42,28,45)(20,31,38,30,35,46,36,47)(25,43,44,39,26,40,34,48);(1,3)(2,8)(4,10)(5,11)(6,13)(7,14)(9,15)(12,16)(17,20)(18,25)(19,30,33,47)(21,35)(22,38)(23,36)(24,39,42,48)(26,27)(28,44)(29,34)(31,37,46,32)(40,41,43,45);(1,4,5,6)(2,9,7,12)(3,10,11,13)(8,15,14,16)(17,20)(18,26)(19,31)(21,35)(22,38)(23,36)(24,40)(25,27)(28,34)(29,44)(30,37)(32,47)(33,46)(39,41)(42,43)(45,48)

Polycyclic presentation:

2^9|a^2=h,b^2=efi,c^2=ehi,d^2=i,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=def,f^a=fh,g^a=gh,c^b=cg,d^b=de,e^b=eh,f^b=fh,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253226);